Stochastic Gradient Descent Langevin Dynamics (SGLD) algorithms, which add noise to the classic gradient descent, are known to improve the training of neural networks in some cases where the neural network is very deep. In this paper we study the possibilities of training acceleration for the numerical resolution of stochastic control problems through gradient descent, where the control is parametrized by a neural network. If the control is applied at many discretization times then solving the stochastic control problem reduces to minimizing the loss of a very deep neural network. We numerically show that Langevin and Layer-Langevin algorithms improve the training on various stochastic control problems like hedging and resource management, and for different choices of gradient descent methods.

We study the convergence rate of the optimal quantization for a probability measure sequence $(\mu_{n})_{n\in\mathbb{N}^{*}}$ on $\mathbb{R}^{d}$ converging in the Wasserstein distance in two aspects: the first one is the convergence rate of optimal quantizer $x^{(n)}\in(\mathbb{R}^{d})^{K}$ of $\mu_{n}$ at level $K$; the other one is the convergence rate of the distortion function valued at $x^{(n)}$, called the “performance” of $x^{(n)}$. Moreover, we also study the mean performance of the optimal quantization for the empirical measure of a distribution $\mu$ with finite second moment but possibly unbounded support. As an application, we show an upper bound with a convergence rate $\mathcal{O}(\frac{\log n}{\sqrt{n}})$ of the mean performance for the empirical measure of the multidimensional normal distribution $\mathcal{N}(m, \Sigma)$ and of distributions with hyper-exponential tails. This extends the results from Biau et al. (2008) obtained for compactly supported distribution. We also derive an upper bound which is sharper in the quantization level $K$ but suboptimal in $n$ by applying results in Fournier and Guillin (2015).

SUMMARY The pricing of American style and multiple exercise options is a very challenging problem in mathematical finance. One usually employs a least squares Monte Carlo approach (Longstaff–Schwartz method) for the evaluation of conditional expectations, which arise in the backward dynamic programming principle for such optimal stopping or stochastic control problems in a Markovian framework. Unfortunately, these least squares MC approaches are rather slow and allow, because of the dependency structure in the backward dynamic programming principle, no parallel implementation neither on the MC level nor on the time layer level of this problem. We therefore present in this paper a quantization method for the computation of the conditional expectations that allows a straightforward parallelization on the MC level. Moreover, we are able to develop for first‐order autoregressive processes a further parallelization in the time domain, which makes use of faster memory structures and therefore maximizes parallel execution. Furthermore, we discuss the generation of random numbers in parallel on a GPGPU architecture, which is the crucial tool for the application of massive parallel computing architectures in mathematical finance. Finally, we present numerical results for a CUDA implementation of these methods. It will turn out that such an implementation leads to an impressive speed‐up compared with a serial CPU implementation. Copyright © 2011 John Wiley & Sons, Ltd.

Shows that the main proofs of the above paper (Yu et al., Trans Neural Networks, vol. 4, no. 2, p. 207-220, 1993) are incomplete and not correct: in fact, the self-organization cannot be achieved if the adaptation parameter satisfies the classical Robins-Monro conditions and Proposition 2 is erroneous. On the other hand, the two-dimensional extension (Theorem 3) is not proved. The main point is that the four classes that the authors consider as stable classes are not stable at all. Some references are finally given.